We study the problem of fitting a description logic (DL) ontology to a given set of positive and negative examples that take the form of an ABox and a Boolean query. While previous work has investigated this problem for the expressive DLs ALC and ALCI, we here focus on the Horn DLs EL and ELI, as well as their extensions with the bottom concept. As the query language, we consider atomic queries (AQs), conjunctive queries (CQs), and unions thereof (UCQs). We provide characterization of the existence of a fitting ontology based on simulations, use them to develop decision procedures, and clarify the exact computational complexity. For AQs, the problem is in PTime for both EL and ELI. For CQs and UCQ, it is Sigma_P^2-complete for EL and ExpTime-complete for ELI. Adding the bottom concept does not change any of these complexities. Interestingly, moving from ALC and ALCI to EL and ELI introduces additional technical challenges rather than simplifying the matter.

We study a fitting problem inspired by ontology-mediated querying: given a collection of positive and negative examples of the form (A, q) with A an ABox and q a query, we seek an ontology O such that A ∪ O entails q for all positive examples (A, q) and A ∪ O does not entail q for all negative examples (A, q). We consider the description logics ALC and ALCI as ontology languages and a range of query languages that includes atomic queries (AQs), conjunctive queries (CQs), and unions thereof (UCQs). For all of the resulting fitting problems, we provide effective characterizations and determine the computational complexity of deciding whether a fitting ontology exists. This problem turns out to be coNP-complete for AQs and full CQs and 2ExpTime-complete for CQs and UCQs. These results hold for both ALC and ALCI.